Physics 5573 E & M I Homework 4: Faraday, Waves, and Electrostatics solved

Description

Problem 1: Faraday Generator

The picture illustrates the concept of a generator based on
Faraday’s law (as most of them are). It consists of an inner
axle of radius 𝑎, outer metal wheel of radius 𝑅, and metal
spokes that connect the two.

The wheel is rotating around
the axle with angular velocity 𝜔, and the whole thing is in
an external magnetic field 𝐵ሬ⃗ that is constant and pointing
out of the page towards you. Due to this motion, an EMF
(voltage) is generated between the axle and the wheel.

A) A common description of this EMF is to use the concept of a “Motional EMF”. This is the
result of a conductor moving through a magnetic field, which causes the electrons in the
conductor to move. For a conductor of length 𝑑𝑥 moving with velocity 𝑣⃗ in a magnetic field 𝐵ሬ⃗:
𝐸𝑀𝐹 ൌ ℇ ൌ “𝐸 𝑑𝑥” ൌ 𝑑𝑥 𝑣⃗ൈ𝐵ሬ⃗

Note: The “𝐸 𝑑𝑥” corresponds to the 𝐸ሬ⃗ ⋅ 𝑑𝑙⃗in Faraday’s Law:
ර 𝐸ሬ⃗ ⋅ 𝑑𝑙⃗ ൌ െ 𝑑
𝑑𝑡ඵ 𝐵ሬ⃗ ⋅ 𝑛ො 𝑑𝑆

For the situation shown, calculate an expression for the total EMF for each spoke in the wheel.
B) As shown in class, the “Motional EMF” is included in Faraday’s law by taking the total time
derivative (rather than partial derivative) of the magnetic flux.

Consider the loop shown in the picture consisting of the dashed lines and one of the spokes. In this loop,
the horizontal dashed line is fixed while the spoke moves. The magnetic flux is changing.

Use Faraday’s law to calculate the EMF around the loop. Show that this EMF is the same as found in Part
A for a spoke, both in magnitude and direction.

C) If you wish to create an EMF ℇ ൌ 5 𝑉 using a magnetic field of 𝐵 ൌ 0.1 𝑇 and a wheel with outer
radius 𝑅 ൌ 0.4 𝑚 and an axle of radius 𝑎 ൌ 0.02 𝑚, how fast must the wheel turn, 𝜔?
𝝎
𝑹
𝒂
𝑩ሬሬ⃗

Problem 2) Conductor in a field

A recent Workshop was about the properties of a
conducting sphere in a constant electric field, several
different approaches to this problem. Here you’ll use a
general solution to Laplace’s equation.

For a problem that has a spherical boundary and is
independent of the azimuthal angle 𝜙, Laplace’s
equation is solved by:
𝜙ሺ𝑟⃗ሻ ൌ ෍൬𝑎௟ 𝑟௟ ൅
𝑏௟
𝑟௟ାଵ ൰ 𝑃௟ሺcos 𝜃ሻ
௟

a) Consider the boundary condition on 𝜙ሺ𝑟⃗ሻ in the limit as |𝑟⃗| → ∞. Show that,
considering this limit, only one term in the sum will be non‐zero.

b) Using the |𝑟⃗| → ∞ limit, solve for one of the remaining coefficients ሺ𝑎௟ or 𝑏௟ሻ.

c) Consider the boundary condition for 𝜙ሺ𝑅ሻ from the workshop, 𝜙ሺ𝑅ሻ ൌ 0. Use this
result to calculate 𝜙ሺ𝑟⃗ሻ for all 𝑟⃗. Show that your result is the same as what was found in
the workshop (and the text).

Problem 3) Cartesian Boundary Conditions

The following is a standard problem in electrostatics, so you can probably find the solution if
you search. Try to do this problem without looking up the solution.

Consider a conducting cube with walls at 𝑥 ൌ 0, 𝑥ൌ𝐿, 𝑦 ൌ 0, 𝑦ൌ𝐿, 𝑧 ൌ 0, 𝑧ൌ𝐿.
The wall at 𝑧ൌ𝐿 has 𝜙ሺ𝑥, 𝑦, 𝑧ൌ𝐿ሻ ൌ 𝑉଴. All other walls of the cube are grounded, 𝜙 ൌ 0.

a) Show that a separable solution of the form:
𝜙ሺ𝑟⃗ሻ ൌ 𝑋ሺ𝑥ሻ 𝑌ሺ𝑦ሻ 𝑍ሺ𝑧ሻ
Is a solution to the Laplace equations. What are the most general functional forms for
𝑋ሺ𝑥ሻ, 𝑌ሺ𝑦ሻ, 𝑍ሺ𝑧ሻ that will solve Laplace? What are the relationships between the
solutions for 𝑋ሺ𝑥ሻ, 𝑌ሺ𝑦ሻ, and 𝑍ሺ𝑧ሻ?

b) Using the boundary conditions on the cube for the walls 𝑥 ൌ 0, 𝑥ൌ𝐿, 𝑦 ൌ 0, 𝑦ൌ𝐿,
what are the possible functions 𝑋௠ሺ𝑥ሻ and 𝑌௡ሺ𝑦ሻ? Explain why these can be indexed by
the integers: 𝑚, 𝑛 ൌ 1, 2, 3, …
𝑥
𝑧
𝑬ሬሬ⃗
𝟎ሺ𝒓ሬ⃗ሻൌ𝑬𝟎 𝒛ො

c) Using the boundary condition for 𝑧 ൌ 0 and the results from (b), what are the
solutions to the function 𝑍௠௡ሺ𝑧ሻ? Be sure to show and explain the dependence of
𝑍ሺ𝑧ሻ on 𝑚, 𝑛, the integers indexing the functions 𝑋௠ሺ𝑥ሻ, 𝑌௡ሺ𝑦ሻ.

d) Write down the general solution to this problem:
𝜙ሺ𝑟⃗ሻ ൌ ෍ 𝑎௠௡ 𝑋௠ሺ𝑥ሻ 𝑌௡ሺ𝑦ሻ 𝑍௠௡ሺ𝑧ሻ
௠,௡
Using the boundary condition for 𝑧ൌ𝐿 and Fourier analysis, determine the coefficients
in this sum, 𝑎௠௡. Show your work.

e) Write a sum that gives the potential at the center of the cube, 𝜙 ቀ௅
ଶ ,
௅
ଶ ,
௅
ଶ
ቁ. You might
be able to simplify this result using the relation:
sinhሺ𝑥ሻ ൌ 2 sinh ቀ
𝑥
2 ቁ cosh ቀ
𝑥
2 ቁ

Show that this sum converges quickly by calculating the value of the first few terms.
Your answers will be 𝑉଴ times a number. You might also determine the ratio of
successive terms in the sum for large m and n.

Problem 4) Dipole Image

Consider a very large (assume infinite) grounded,
conducting sheet lying in the 𝑧 ൌ 0 plane. A polar
molecule is on the z‐axis at 𝑟⃗
௣ ൌ 𝑑 𝑧̂. You can model
the molecule as a point dipole, 𝑝⃗, with the dipole
moment in the x‐z plane at an angle 𝜃 as shown.

a) Determine an image that can be used to solve for the electric potential and field for
this situation for all 𝑟⃗ with 𝑧 ൒ 0. Show that your image gives the correct boundary
condition for this problem, 𝜙ሺ𝑥, 𝑧 ൌ 0ሻ ൌ 0 for any 𝑥.

b) Determine an expression for the potential 𝜙ሺ𝑥, 𝑧ሻ for any 𝑥 and 𝑧 ൐ 0.

c) Determine the electric field everywhere on the conducting sheet (meaning for points
𝑧 → 0, 𝑧 ൐ 0). You can do this by either taking a derivative of your result from (b) or
summing the fields due to the molecule (dipole) and the image (or both, of course).
d) Consider the two cases, 𝜃 ൌ 0 and 𝜃 ൌ గ
ଶ
.

For each of these angles, calculate:
i) The surface charge density on the conducting sheet for any 𝑥.
ii) The force on the molecule (dipole) due to the conducting sheet.
𝑥
𝑧
𝒑ሬሬ⃗
𝜃
𝜙ሺ𝑧 ൌ 0ሻ ൌ 0
𝑟⃗
௣ ൌ 𝑑 𝑧̂